Prime numbers with a certain extremal type property

Abstract

The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points (e_k,π(e_k))₁^∞. The elements of the sequence (e_k)₁^∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (e_k)₁^∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then e_k+1/e_k=1. The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.